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August  2007, 17(3): 481-500. doi: 10.3934/dcds.2007.17.481

On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system

Vladimir V. Chepyzhov 1, , E. S. Titi 2, and  Mark I. Vishik 1,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127994, GSP-4, Russian Federation, Russian Federation
2. 

Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States

Received  January 2006 Revised  August 2006 Published  December 2006

We study the relations between the global dynamics of the 3D Leray-$\alpha $ model and the 3D Navier-Stokes system. We prove that time shifts of bounded sets of solutions of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and $ \alpha $ approaches zero. In particular, we show that the trajectory attractor of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system when $\alpha \rightarrow 0\+.$
Keywords: trajectory attractors., 3D Navier--Stokes system, 3D Leray-$\alpha $ model.
Mathematics Subject Classification: Primary: 35Q30, 37L30; Secondary: 76D03, 76F20, 76F55, 76F6.
Citation: Vladimir V. Chepyzhov, E. S. Titi, Mark I. Vishik. On the convergence of solutions of the Leray-$\alpha $ model to the trajectory attractor of the 3D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 481-500. doi: 10.3934/dcds.2007.17.481
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